Mass as Gravitational Mass

Recent posts discuss the concept of mass concluding that gravitational mass appears to be primordial from which inertial mass is derived. The idea is to start with a Universe defined in terms of a gravitational potential $\phi (x,t)$ depending on a Euclidean space coordinate $x$ and a time coordinate $t$ as created in 3 steps:

1. The gravitational potential creates a mass density $\rho (x,t)=\Delta\phi (x,t)$, where $\Delta$ is the Laplacian differential operator acting instantly as a local operation in space. 

2. Mass is subject to free fall according to Newton's equations of motion subject to gravitational force $\nabla\phi (x,t)$. 

3. Redistribution of mass under free fall gives feed-back to gravitational potential.

We thus find a Newtonian Universe determined by gravitational free fall connecting motion of mass in space to gravitational force and so equating inertial mass to gravitational mass. This is a large scale in a sense complete Universe with a precise simple mathematical description, which can be complemented by electromagnetics without interference with gravitation on both macro-scale and atomic micro-scale into the Universe we can see and experience.  

Modern text-books tell another story with mass appearing as (i) gravitational mass, (ii) inertial mass, (iii) rest mass and (iv) Higg's mass, all of different nature, which is is very confusing and lacks reason.  

It seems to be more reasonable to define mass as gravitational mass, which is the operational definition according to SI 2019 standard of units, and then connect other expressions of mass to this standard. 

On the other hand, the formal presence of mass $m$ in the coefficient $\frac{h^2}{2m}$ of the Laplacian in Schrödinger's Equation SE , with the mass of a proton 1836 times that of an electron, is not connected to gravitation and free fall motion.  Instead $m$ here serves as a parameter to determine spatial size, with thus a proton having smaller size than an electron in an atom.  

Different concepts of mass in modern physics.

 

Mass of Many Different Forms and Origins

I have been led to the idea that there is in Newtonian mechanics only one form of mass, gravitational mass, which is derived from a gravitational potential as primordial, wit inertial mass = gravitational mass as a consequence of universality of free fall and conservation of energy. This is a clear picture, which can be understood by everybody, and which has shown to work excellent in practice. 

This is not the idea of modern physics, which plays with several different forms of mass:  

• gravitational mass
• inertial mass
• rest mass (relativistic)
• Higgs mass. 

with different origins:

• gravitation
• Higgs mechanism and strong force
• relativity theory
• Higgs mechanism.

No doubt this shows a very complex picture and the questions pile up: Why so many different forms with different origins and why then is an Equivalence Principle adopted stating that inertial mass = gravitational? 

• Warum es einfach machen, wenn mann es so schön kompliziert machen kann?

  

Inertial Mass = Gravitational Mass?

This is a continuation of previous posts on operational definition in SI 2019 of mass as gravitational mass. 

In Newton's mechanics inertial mass is exactly equal to gravitational mass as an expression of both (i) universality of free fall and (ii) conservation of energy.  A Universe without (i) and (ii) cannot exist.

In Einstein's mechanics this truly fundamental equality is no longer guaranteed by (i) and (ii), but has to be added as an independent Equivalence Principle EP, which has to be supported by experimental evidence. Accordingly major efforts have been made to find ever more precise experimental confirmation, where the current precision is $10^{-15}$, while new experiments are receiving funding to reach even better precision: 

✅ Best Precision to Date:

• Experiment: MICROSCOPE (CNES, ESA)

• Method: Differential accelerometry in space (free-falling test masses of different materials)

• Result: Difference in acceleration between test masses of different compositions was less than 2 parts in $10^{15}$.

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🚀 Future Target Precisions:

1. Galileo Galilei (GG) satellite – proposed

   • Target: $10^{-17}$

2. STE-QUEST (atom interferometry in space) – proposed

   • Target: $10^{-17}$ (depending on mission configuration)

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To a classical physicist these experiments are similar to precise testing of the validity of the equality 1+1 = 2. 

Einstein presented his Special Theory of Relativity SR in 1905 in a desperate effort to get a university position, and followed up in 1915 with his General Theory of Relativity GR in a desperate attempt to keep the position he managed to get in 1909 under severe criticism of SR. 

Einstein thus took on the role of challenging Newton's mechanics by connecting it to propagation of light which was not mechanics, and by assuming EP as a fundamental postulate of GR as a step away from Newton's mechanics.

Despite severe criticism of both SR and little acceptance of GR before his death in 1955, today Einstein is the undisputed hero of modern physics as the man who showed that Newton was wrong and so opened the door to all sorts of new physics desperately needed after the success of the atom bomb in 1945 had faded.   

How to Measure/Define Mass

This is a clarification of the previous post showing gravitational force without need of force carrier.

An instrumentalist definition of a certain quantity like mass depends on how the quantity is measured by a certain specified instrument. 

According to the SI 2019 system of units, mass $m$ is determined by a Kibble balance from gravitational force $mg$ measured as electromagnetic force, where $g$ is the local gravitational constant. 

We conclude that according to SI 2019 mass is gravitational mass as reaction to gravitational force. In other words, gravitational force as gradient of a gravitational potential is primary from which mass is derived as secondary as measured by a Kibble balance. 

This suggests that the relation between mass density $\rho (x)$ and gravitational potential $\phi (x)$ should be viewed to have the form, with $x$ a Euclidean 3d space coordinate and $t$ a time coordinate: 

• $\rho (x,t) = \Delta\phi (x,t)$           (1)

with $\phi (x)$ given and $\rho (x)$ derived by application of the differential operator $\Delta$ acting locally, rather than the standard form: 

• $\Delta\phi (x,t)=\rho (x,t)$,             (2)

with $\rho (x)$ given and $\phi (x)$ derived as solution to Poisson's equation as a differential equation.

We can view (1) as local instant action by differentiation, while (2) requires instant action at distance as integration/summation. 

By adopting (1) as the true connection between gravitational mass and gravitational potential, which fits with SI 2019 and is explored in many posts, we can thus circumvent to need of instant action at distance, which has been viewed as a stumbling stone for Newtonian mechanics, motivating Einstein's relativistic mechanics coming with many new difficulties.

Adopting (1) we thus (i) adhere to SI 2019, (ii) circumvent instant action at distance and (iii) do not need Einstein,  as a hat trick. 

In Newtonian mechanics inertial mass is defined as gravitational mass and their equivalence is not an assumption or Principle as in Einstein's mechanics. 

In the Standard Model, mass is not defined as gravitational mass as in SI 2019, but as an intrinsic quality/property carried by matter, which is formed in a very strange way by the Higg's mechanism. 

Altogether, mass as gravitational mass, appears as a very useful concept. It makes sense to define mass as gravitational mass, because gravitation is present virtually everywhere. A special feature is that free fall as motion under gravitational force, without presence of forces of other nature, is independent of the size of the mass, which defines mass as a form of sensitivity to gravitation which is additive: The mass of two particles of the same kind is twice that of one particle. 

But this is not how a modern physicist approaches the problem of defining what mass is and how to measure it. A modern physicist is trained to believe that mass according to the Standard Model is defined by QFT through the Higg's mechanism, and then connected to gravitation by General Relativity GR. The trouble is the QFT and GR are incompatible and so the mass problem is a mess problem. 

Force Carriers of Modern Physics without Meaning

 The Standard Model SM as the crown jewel of modern physics divides the World into:

• matter particles (fermions)
• force carriers (bosons) of different forms:
• photons for electromagnetic force
• W and Z bosons for weak nuclear force
• gluons for strong nuclear force 
• (gravitons for gravitational force, hypothetical not detected)  

A major short-coming is that SM does not describe gravitation, because the graviton as force carrier has not been identified/detected, while the other force carriers are claimed to have been detected in particle accelerators. 

ChatGPT admits that the whole idea of force carrier or mediator of force by exchange of bosons, may be wrong in the sense that this is not how physics really works. The fact that the idea does not work for gravitation, may indicate that it is an illusion that it works for the electromagnetic and weak/strong forces. Another troubling aspect is that mediation by force carriers come with questions about finite speed of delivery, since infinite speed is hard to imagine. 

Classical physics does not include any concept of force carrier, because it has no role to play. Forces in classical physics can be viewed to come in two forms:

• Potential forces as spatial gradients of potentials instantly delivered locally. Gravitational and electro-magnetic forces are derived this way from electro-magnetic and gravitational potentials. 
• Contact forces also instantly delivered locally in space. Elastic forces in an elastic solid have this form. 

These forces need no force carriers since forces act locally in space without time delay, and so there is no issue with action at distance asking for some form of force carrier. It is thus possible to view the apparent instant action at distance of gravitational force as a misconception since the force is in fact delivered locally by a potential with instant delivery. This is the theme of many posts.

It is thus possible to describe electro-magnetic and gravitational forces without force carriers and then circumvent the apparently unsolvable problem of instant action at distance deeply troubling SM. 

RealNucleus presents a model of an atomic nucleus in terms of only electro-magnetic force. 

Summary:

The fundamental component of SM of force carrier raises several questions:

• How is the force carried and by what?
• How quick is the delivery of force over distance?
• Why is there no force carrier for gravitation?
• How is electro-magnetic force carried by photons? 

Coupled with the fact that there may be no need of force carrier at all, since force is always delivered locally instantly, like no need of surface mail when there is instant email, it may seem better to eliminate it from the discussion altogether as only a novelty without lasting value invented by a modern physics in desperate need to deliver new physics. 

18th century Stangenkunst (system of linked rods) as carrier of mechanical force over long distance.

PS ChatGPT informs us: 

• The concept of force carriers was introduced out of necessity when physicists tried to extend classical ideas to the quantum world. It arose to explain how particles interact in a way consistent with both quantum mechanics and special relativity.
• The carriers of the electromagnetic force are virtual photons which are internal lines in Feynman diagrams. The were introduced to get around the contradiction between instant-action-at-distance and special relativity, which however does not arise since instant-action-at-distance has no role to play as we saw above.

There is thus no need of any virtual photons and so we do not have to worry about the physical meaning of lines in a Feynman diagram. The basic idea of RealQM is that there is no conceptual difference between macroscopic and microscopic physics, that is that the quantum world is not fundamentally different from the world we live in. If this is indeed the way things are, it will make the atomic world easier to understand and then to manipulate.   

Modern Physics as Extreme Physics in Simple Geometry

Modern physics can be described as the physics of the extreme, while classical physics concerns the normal non-extreme. The atomic bomb is an ultimate expression the extreme. Powerful particle accelerators force subatomic particles such as protons, electrons and heavy nuclei to smash into each other at extremely high speeds creating a spray of other particles collected in detectors, with the objective of discovering the inner structure of the particles which are smashing. It is similar to seeking to discover the inner structure of a Swiss clock or human cell by smashing it with a powerful hammer.

Einstein's Equations EE are presented as a more accurate/fundamental model of gravitation than Newton's Equations NE, and the evidence is picked from cases of very strong gravitation such as mergers of black holes, which can be described as extreme cases with simple geometry allowing solutions to EE equations to be determined, more or less. 

However, evidence that EE is a more accurate model of gravitation than NE for normal cases of classical physics such as planetary systems, which can be described as normal cases with complex geometry for which NE works fine, is missing because in such cases EE are uncomputable and thus cannot be inspected and compared with NE/observation. 

The argument appears to be that if a model works in an extreme case, it should work also in a non-extreme case, but it is not really valid here because the extreme case has simple geometry allowing solutions of EE to be constructed, more or less, while the normal case has complex geometry, which can be handled by NE. 

EE is presented as one of the two major achievements of modern physics as a more accurate/fundamental model of gravitation than NE, but this cannot be demonstrated in the vast majority of normal cases, because EE is uncomputable in complex geometry. 

The other major achievement of modern physics is quantum mechanics based on Schrödinger's equation SE with extensions as QED and QCD underlying the Standard Model of fundamental particles of spatial scale down to $10^{-18}$ m. But SE is uncomputable for normal systems with complex geometry just as EE, which has led physicists to shift focus to string theory of scale $10^{-32}$ m as an expression of ultimate extreme physics in simple geometry. 

Modern physics thus has come to concentrate on extreme physics in simple geometry, in an attempt to distance itself from classical physics of the normal in complex geometry, which covers the majority of cases. No wonder that modern physics is in a state of crisis.  

Extreme physics in simple geometry

  

Human Protein Atlas: Normal physics in complex geometry

Einstein's Equations vs Precession of Mercury

When Einstein in 1916 presented a new model of gravitation in the form of  Einstein's equations, he desperately needed evidence that his model was better than Newton's. He found this in a back-of-the-envelope computation adding precisely the missing 43 arcseconds in the extra precession of the perihelion of Mercury (slight perturbation of the rotation of the elliptical orbit of Mercury of 5600 arcseconds per century from other planets), to make the prediction by Newton's model of 531 arcseconds/century made by Le Verrier in 1859 to fit exactly with the observed 574 arcseconds/century. 

Einstein thus started from the known results of 531 by Newton and observed 574 and miraculously obtained exactly the missing 43 by a very simple computation which he claimed captured the difference between his and Newton's model.  

If we ask chatGPT about this apparent miracle we get the following information:

1. Le Verrier's 531 has been confirmed to within 1 arcsecond by modern high tech computation and precise data. This is in itself a miracle. No improvement since 1859!
2. It is impossible to directly compute the observed 574 by solving Einstein's equations including in particular the other planets, because the equations are impossible to solve both analytically and computationally. 
3. Computation always start from 531 with Newton's equations and 571 is obtained as a correction of 43 claimed to be based on Einstein's equations, even if they are impossible to solve. This is also a miracle.

To get perspective on these miracles as prime evidence that Einstein's equations are more precise than Newton's, let us note 

• 1 arcsecond/century corresponds to a relative accuracy of about $2\times 10^{-9}$ per revolution of Mercury (415 revolutions/century) if errors add up linearly. 
• Data errors cannot be expected to cancel and so may well add up linearly. 
• A precision of $2\times 10^{-9}$ is thus needed in data such as planet masses, gravitational constant G, initial data, composition of the Sun, tidal motions, other celestial objects than planets, to get the precession right to say 1 arcsecond over 100 year. 
• The gravitational constant is known to at best 6 decimal places, other data with less. 

The conclusion that the desired precision cannot be reached, cannot be avoided. There is factor of 1000 between required and available precision. Yet, the computation of the precession of the perihelion of Mercury by Einstein is still presented as prime evidence that Einstein's equations describe gravitation better than Newton and thus as an undeniable major victory of modern physics over classical physics. 

Check if this is confirmed by chatGPT after careful reading of the literature!

But at his deathbed Einstein begged: "Newton, forgive me!"

Politics vs Modern Physics

A common tactic in politics practiced by parties and governments is to construct or frame a problem and then position themselves as the best (or only) solution to the problem as agenda-setting, problem-framing and solution-ownership. The problem typically a fictitious invented problem that does not need to be resolved as a no-problem. 

The problem could be imminent invasion of a NATO country by Russia, like Sweden, which can be prevented/solved by boosting Swedish defense industry with tax payer money. 

The same tactic has been used to gain tax payer support to modern Big Physics in the form of particle accelerators and massive support to departments of theoretical/fundamental physics. 

It was Boltzmann who took the first step along this profitable road at the end of the 19th century by inventing statistical mechanics as a new form different from classical mechanics, to solve the constructed problem of explaining why time is moving only forward, which is a no-problem in physics following its own dynamics.  

Planck took the next step in 1900 coming up with a concept of quanta as small chunks of energy to solve the problem of ultra-violet catastrophe of black-body radiation giving all bodies the capacity of radiating an infinity of energy, as a no-problem in real physics.  

In 1905 Einstein followed by inventing the Special Theory of Relativity SR starting from a proclaimed difficulty for two (human) observers separated in space to decide if two events are simultaneous in time. This is a constructed purely observer dependent problem, which does not come up in real physics without observers. Physics happens here and now and simultaneity of events (widely) separated in space is a no-problem. 

In the 1920s modern physics took a big leap inventing quantum mechanics with inspiration from Planck to solve the problem of the Bohr model of the Hydrogen atom. The solution introduced a whole catalog of new problems asking for resolution including wave-particle duality, collapse of the wave function, Heisenberg Uncertainty, Pauli Exclusion, anti-symmetry, interpretation of the wave function...The next big leap took place in the 1960s inventing the Standard Model to solve the problem of stability of atomic nuclei and then coming with another catalog new problems including strong force and weak force as different from familiar Coulomb force, and nuclear force as residual of strong force.  

In the 1970s String Theory was invented to solve perceived problems of the Standard Model. Today it is clear it did not work.

We see a consistent development of modern physics since 1900 as ad hoc inventions to solve perceived problems of classical physics by going outside classical physics, which were either no-problems or problems which could have been solved within classical physics (see books listed on this blog),  but then produced a host of new problems asking for more tax money. 

This machine has served very well for 100 years, boosted by the triumph of nuclear bombs, but it seems that public support is now failing as an expression of a crisis of modern physics. 

What Is an Electron? RealQM vs StdQM.

Electrons are described by wave functions as solutions to Schrödinger's Equation SE as the basic mathematical model of the quantum mechanics of atoms and molecules which comes in two forms: Standard Quantum Mechanics StdQM and Real Quantum Mechanics RealQM:

• Wave functions of StdQM for a system with $N$ electrons have global supports and depend on $N$ 3d spatial coordinates, altogether $3N$ spatial coordinates. Wave functions are computable only for very small $N$ since computational complexity grows exponentially with $N$.
• Wave functions of RealQM are sums of one-electron wave functions with non-overlapping supports depending on the same 3d spatial coordinate, and meet with continuity at a free boundary between supports. Wave functions are computable for all $N$, since computational complexity grows linearly with $N$. 

We now consider in more detail electrons according to RealQM. Each electron is described as a charge density $\psi (x)^2$ of a real-valued wave function $\psi (x)$ with support (non-zero value) in a certain region $\Omega$ in space with boundary $\Gamma$. The electron charge density does not have to vanish on $\Gamma$. Electrons sharing a common boundary piece meet with continuity as a free boundary condition. This allows the kinetic energy of an electron measured by $\vert\nabla\psi (x)\vert^2$ to be small even if the size of $\Omega$ is very small, which is not possible if $\psi (x)$ is forced to vanish on $\Gamma$. In other words, in the presence of other electrons, an electron can have small support and small kinetic energy, which means that it can circumvent the dictate of Heisenberg's Uncertainty Principle.

RealQM describes protons in the same way as electrons, with only a shift of sign of the charge.  

RealQM allows electrons to appear in two different forms in atoms and atomic nuclei as: 

• Charge densities with large support in atoms around a nucleus of vanishing size, meeting the nucleus freely (RealAtom). 
• Charge densities with very small support in kernels of nuclei surrounded by protons meeting proton charge densities with continuity, making sense since the size of the kernel is not small compared to the nucleus (RealNucleus). 

RealQM thus offers a complete model of an atom + nucleus in terms of: 

• A Schrödinger equation for a collection of positive and negative charge densities with non-overlapping supports interacting by Coulomb potentials, 
• The large difference in mass between proton and electrons allows the electron to serve a double role with presence both outside and inside the nucleus.  

RealQM is non-relativistic with motivation from the fact that there is no charge density motion in neither atom nor nucleus at all, and then certainly not at any relativistic speed. This is fundamentally different in StdQM where inner core electrons of heavy elements like Gold are claimed to move at half the speed of light by a purely formal argument connecting kinetic energy to velocity. Electrons can in giant particle accelerators be accelerated to relativistic speeds with massive input of energy, but its is difficult to fathom that the same thing happens in an atom of Gold...

Summary:

• RealQM describes a collection of $N$ electrons as a charge densities with non-overlapping local supports in 3 space dimensions meeting with continuity and interacting by Coulomb potentials. RealQM is computable for all $N$.
• StdQM the collection as overlapping charge densities with global support in $3N$ space dimensions. StdQM is computable only for very small $N$. 
• RealNucleus describes a nucleus as a collection of non-overlapping electron and proton charge densities meeting with continuity and interacting by Coulomb potentials.
• The Standard Model of StdQM describes a nucleus as a collection of protons and neutrons interacting by a residual of a strong force.   

 

   

Modern Physics as Poker Game

Modern physics based on relativity theory and quantum mechanics has followed the pattern set by Einstein to continue to raise the bet without showing the cards, as in a no-limit-poker game with lousy cards, eventually forcing the other players to fold: 

1. When Einstein's Special Theory of Relativity SR (1905) was questioned, Einstein raised the bet to the General Theory of Relativity GR (1916).
2. When GR was questioned, Einstein raised the bet from scales of classical physics, to scales of the Universe including black holes which nobody dared to call. 

This showed to work fine and so the same strategy was adopted by leading physicists as concerns quantum mechanics:

1. When the physical meaning of Schrödinger's Equation SE (1926) for atoms of size $10^{-10}$ m as the first version of quantum mechanics without relativity theory, was questioned and no answers could  be given, it was replaced by Dirac's Equation DE (1930) including SR describing an electron.
2. When DE for the electron was questioned, the Standard Model SM (1960s) was developed as a theory of atomic nuclei of size $10^{-15}$ m built from the fundamental particles of protons and neutrons made up of quarks interacting by force carriers named gluons. 
3. When SM was questioned as an ad hoc model of a nucleus, String Theory ST (1980s...) was developed as an ultimate fundamental theory on a scale ($10^{-32}$ m ) which could not be called.   

ST/GR ars now being questioned, but the bet cannot be raised another time since the ultimate large and small scales have been reached. It is now time for physicists to call and to help there is a new card to play in the form of RealQM. Anyone with this card can make a call on ST/GR.